Factored and General Forms of Polynomial and Rational Functions; Pascal's Triangle and the Binomial Theorem

Factored Form vs General Form

Polynomial Functions

Polynomial functions can be written in different forms, each providing unique insights into their properties and behavior.

• Factored Form: Expresses the polynomial as a product of its linear (or irreducible) factors.

  • General format: $f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}\ldots(x - r_n)^{m_n}$

  • Zeros: The values $r_1, r_2, \ldots, r_n$ are the zeros (roots) of the function.

  • Multiplicity: The exponent $m_i$ indicates the multiplicity of each zero.

  • Application: Easy to find zeros and their multiplicities.

• Standard Form: Expands the polynomial as a sum of terms in descending powers of $x$.

  • General format: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

  • Degree: The highest power of $x$ (i.e., $n$) is the degree of the polynomial.

  • Y-intercept: The constant term $a_0$ gives the y-intercept.

  • Application: Useful for identifying end behavior and y-intercept.

Example #1

• Given: $f(x) = 2(x + 4)(x - 2)^2 - 8x$

• Zeros: Set each factor equal to zero: $x = -4$, $x = 2$ (with multiplicity 2).

• Degree: Add the exponents: $1 + 2 = 3$ (cubic polynomial).

• End Behavior: Since the leading coefficient is positive and degree is odd, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

• Y-intercept: Substitute $x = 0$ into the function.

Rational Functions

Rational functions are ratios of two polynomials and can also be written in factored or general form.

• Factored Form: $f(x) = \frac{(x - a_1)(x - a_2)\ldots}{(x - b_1)(x - b_2)\ldots}$

• General Form: $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials.

• Domain: All real numbers except where $Q(x) = 0$ (denominator is zero).

• Zeros: Values of $x$ where $P(x) = 0$ (numerator is zero).

• Vertical Asymptotes: Values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator.

• Holes: Occur at values of $x$ where both $P(x)$ and $Q(x)$ are zero (common factors).

Example #2

• Given: $f(x) = \frac{(x-3)(x+5)}{(x+3)(x-1)}$

• Zeros: $x = 3$, $x = -5$

• Vertical Asymptotes: $x = -3$, $x = 1$

• Horizontal Asymptote: Since degrees of numerator and denominator are equal, horizontal asymptote at $y = 1$ (ratio of leading coefficients).

• Holes: None (no common factors in numerator and denominator).

Pascal's Triangle

Structure and Use

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. It is used to find coefficients in binomial expansions.

• Each row corresponds to the coefficients of the expanded form of $(a + b)^n$.

• Row $n$ contains $n+1$ entries.

• For example, the 4th row (1, 4, 6, 4, 1) gives the coefficients for $(a + b)^4$.

How to Construct Pascal's Triangle

• Start with 1 at the top.

• Each subsequent row starts and ends with 1.

• Each interior number is the sum of the two numbers above it.

Application to Binomial Expansion

• To expand $(a + b)^n$, use the $n$th row of Pascal's Triangle for coefficients.

• Each term: $C_k a^{n-k}b^k$, where $C_k$ is the $k$th entry in the $n$th row.

• Example: $(x + 5)^4 = 1x^4 + 4x^3(5) + 6x^2(25) + 4x(125) + 1(625)$

Binomial Theorem

Statement and Formula

The Binomial Theorem provides a formula for expanding powers of binomials.

• General Formula:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

• $\binom{n}{k}$ is the binomial coefficient, equal to the $k$th entry in the $n$th row of Pascal's Triangle.

• Example: Expand $(x + 2)^3$:

$(x + 2)^3 = 1x^3 + 3x^2(2) + 3x(4) + 1(8) = x^3 + 6x^2 + 12x + 8$

• Application: Used for quick expansion of binomials without direct multiplication.

Summary Table: Key Differences Between Forms

Form	Polynomial Function	Rational Function
Factored Form	Shows zeros and multiplicity	Shows zeros, domain restrictions, and possible holes
General/Standard Form	Shows degree, end behavior, y-intercept	Shows end behavior, horizontal asymptotes

Additional info: The notes also include color-coded annotations for clarity, such as reminders to check for holes in rational functions and to use Pascal's Triangle for binomial expansions. The examples provided illustrate how to apply these concepts to specific functions.